equation of motion
equation of motion, mathematical formula that describes the position, velocity, or acceleration of a body relative to a given frame of reference. Newton’s second law, which states that the force F acting on a body is equal to the mass m of the body multiplied by the acceleration a of its centre of mass, F = ma, is the basic equation of motion in classical mechanics. If the force acting on a body is known as a function of time, the velocity and position of the body as functions of time can, theoretically, be derived from Newton’s equation by a process known as integration.
For example, a falling body accelerates at a constant rate, g. Acceleration is the rate of change of velocity with respect to time, so that by integration the velocity v in terms of time t is given by v = gt. If one graphs acceleration versus time, the velocity v would be the area under the curve. Velocity is the time rate of change of position S, and, consequently, integration of the velocity equation yields S = 1/2gt2. Similarly, if one graphs velocity versus time, the area under the curve gives the change in position, and the derivative of the velocity, or the slope of the velocity curve would be the acceleration (in this case, g). If a body is moving with a uniform velocity, the slope of the velocity curve would be zero.
If the force acting on a body is specified as a function of position or velocity, the integration of Newton’s equation may be more difficult. When a body is constrained to move in a specified manner on a fixed path, it may be possible to derive the position-time equation; from this equation the velocity-time and acceleration-time equations can, theoretically, be obtained by differentiation.
